After we get the expression for the dynamic matrix, we can calculate
the eigenvalues and eigenvectors of the effective Hamiltonian. Inserting
the eigenvalues, eigenvectors and the \(D\)  matrix to the formula
Eq. (REF ), we are able to compute the phonon Hall conductivity.
As is well known, in quantum Hall effect for
electrons, the Hall conductivity is just the Chern number in units
of \(e^2/h\)  (\(h\)  is the Planck constant); thus with the varying of
magnetic field, the abrupt change of Chern numbers directly induces
the obvious discontinuity of the Hall conductivity. However, for the
PHE, there is an extra weight of \((\omega _\sigma +\omega _{\sigma ^{\prime }})^2\)  in Eq. (REF ), which can not be moved out from the
summation. As a consequence, the change of phonon Hall conductivity
is smoothened at the critical magnetic field. However, in the study on the
PHE in the honeycomb lattice system [1]}, from the first
derivative of phonon Hall conductivity with respect to the magnetic
field \(h\) , at the critical point \(h_c\) , we still can observe the
divergence (singularity) of \(d\kappa _{xy}/dh\) , where the phase
transition occurs corresponding to the sudden change of the Chern
numbers. Can such mechanism be applied for the kagome lattice system?
In the following, we give a detailed discussion on it.
